Final answer:
To find the value of constant c, we need to evaluate f/g'(2) = 4, where f = 2x^3c and g = x^2. Substituting the values and solving for c, we find that c = 8/x^3.
Step-by-step explanation:
To find the value of constant c, we need to evaluate f/g'(2) = 4, where f = 2x^3c and g = x^2.
To find g'(x), we need to differentiate g with respect to x. The derivative of g(x) = x^2 is g'(x) = 2x.
Substituting the values into f/g'(2) = 4, we get (2x^3c)/(2(2)) = 4. Simplifying this equation, we have x^3c/2 = 4. Solving for c, we can multiply both sides by 2 and divide by x^3 to get c = 8/x^3.
To find the value of the constant c given that the ratio of the function f to the derivative of the function g at x=2 equals 4, where f = 2x³ + c and g = x², we first need to find the derivative of g, denoted as g′. The derivative of g with respect to x is g′(x) = 2x. At x = 2, g′(2) = 2(2) = 4.
We then calculate the value of f at x = 2: f(2) = 2(2)^3 + c = 16 + c. Using the given ratio f/g′(2) = 4, we set up the equation (16 + c)/4 = 4. Solving this equation for c gives c = 4 * 4 - 16 = 0. Therefore, the value of the constant c is 0.